Hamiltonian rotating-wave approximation

For a three-level system the Hamiltonian in the interaction picture //, in Rotating Wave Approximation is given in matrix representation by... 

As a consequence, using the Bosons description and performing the rotating wave approximation lead us to write the effective Hamiltonians (102) and (103) describing the H-bond bridge coupled to the thermal bath as follows ... 

We now consider this scenario in detail by first writing the CPT (radiatiod + matter) Hamiltonian in the rotating wave approximation as... 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

The most general dressed Hamiltonian in the rotating wave approximation for these processes reads [69]... 

We consider a two-level atom with excited and ground states e) and g) when in a photonic crystal coupled to the field of a discrete (or defect) mode and to the photonic band structure in the vacuum state. The hamiltonian of the system in the rotating-wave approximation assumes the form [Kofman 1994]... 

Here we discuss in detail a model for measurement-induced decay modification in a multilevel system. The system with energies frwn, 1 < n < N, is coupled to a zero-temperature bath of harmonic oscillators with frequencies uj. The corresponding Hamiltonian, in the rotating-wave approximation, is... 

The exceptionally short lifetime prompted a re-examination of electron-transfer theory in the presence of an electronic continuum as the bath —in place of the conventional nuclear continuum. This analysis used the starting Hamiltonian, within the rotating wave approximation... 

The total Hamiltonian describing the energies of the systems, electromagnetic field and interactions, in the electric dipole and RWA (rotating-wave approximation) approximations [21], is composed of four terms... 

Another model of the detector, which has only two energy levels, has been considered [188]. The most significant features can be described, in the rotating-wave approximation (RWA), in the framework of the following generalization of the Jaynes-Cummings (JC) Hamiltonian ... 

In order to solve for c(7) we consider the dynamics in the rotating-wave approxima J tion. One convenient way to do this is to write the Hamiltonian directly in this J approximation, and neglect off-resonance terms. This corresponds to a molecule- ... 

In the interaction picture, with the rotating-wave and electric-dipole approximations, the interaction Hamiltonian can be written as... 

Q3. The interaction Hamiltonian in the rotating wave and dipole moment approximations for the three-level system is... 

Several decoupling approximations have been developed to simplify treatments where many rotational channels are coupled to begin with. A number of calculations have used orbital-rotational decoupling in the body-fixed frame, an il-dominant decoupling, and helicity decoupling. Several of these approaches have been recast in terms of effective Hamiltonians. Other decoupling treatments have extended the distorted-wave approximation by means of exponential operators or with optical potentials. 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

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